Hibachi are technically a traditional Japanese device used for heating one's house. They are a basic, heat-proof container that holds charcoal.
The cooking devices that many people refer to as "hibachi" are what the Japanese would call "shichirin":
I'm guessing that the term "hibachi" was popularized in North America because "shichirin" can be hard to pronounce for Anglophones.
Somewhere along the way, primarily in North America, the term "hibachi" also started to be used to refer to teppanyaki:
I'm not sure when or why this started; perhaps it has something to do with the fact that Banihana confusingly refers to their teppanyaki restaurants as "hibachi-style".
Among these, teppanyaki is most similar to Mongolian barbecue, in which meat is cooked on large, round, cast iron griddles:
(Images taken from Wikipedia.)
If you were to actually go to a Japanese restaurant and cook your own food over a shichirin, it would likely be referred to as "yakiniku", which is believed to have some origins in Korean barbecue.
Whereas teppanyaki has been a traditional Japanese cooking method for a long time, "Mongolian barbecue" was developed in the 1970s in Taipei, Taiwan. During that time, Japanese Teppanyaki was very popular in Taiwan, so many people speculate that was actually the inspiration for Mongolian barbecue. There are also some similarities between the Japanese dish "jingisukan" and Mongolian barbecue, however, jingisukan predates Mongolian barbecue.